On Riesz subsets of Abelian discrete groups
type: Item
instance of: scholarly article
title: On Riesz subsets of Abelian discrete groups
zbMATH Open document ID: 0661.43003
DOI: 10.1007/BF02772575
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let G be a compact abelian group, and \(\Gamma\) its dual. A subset \(\Lambda\) of \(\Gamma\) is a Riesz set if every bounded measure whose Fourier coefficients vanish off \(\Lambda\) is absolutely continuous with respect to the Haar measure of G. The theorem of F. and M. Riesz (1916) is that the set of positive integers is a Riesz subset of the integers. The paper here is an encyclopedic collection of results on Riesz sets, some old, some new. For example, it includes a proof of the Mooney-Havin theorem that the quotient \(L^ 1(T)/H^ 1(D)\) is weakly sequentially complete. Let \(\sigma\) be the Haar measure of G. Then f is in weak \(L^ 1\) if the products \(t\sigma (| f| >t)\), \(t>0\), are bounded. A subspace of \(L^ 1\) is nicely placed if its unit ball is closed in weak \(L^ 1\), while a subset \(\Lambda\) of the dual \(\Gamma\) is nicely placed if the space \(L^ 1_{\Lambda}\) of functions in \(L^ 1(G)\) whose Fourier coefficients vanish off \(\Lambda\) is nicely placed. Finally, \(\Lambda\) is a Shapiro set if every subset of \(\Lambda\) is nicely placed. Joel Shapiro (1978) proved that the set of positive integers is a Shapiro subset of the integers. The principal result of the paper here is that if \(\Lambda\) is a Shapiro set, then \(\Lambda\) is a Riesz set, and the quotient \(L^ 1/L^ 1_{\Lambda}\) is weakly sequentially complete. The paper includes several striking arithmetic results. For example, the set of squares of integers is closed in the Bohr compactification of the integers.
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 4080120
zbMATH Keywords: compact abelian group
zbMATH Keywords: bounded measure
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Haar measure
zbMATH Keywords: Riesz sets
zbMATH Keywords: weakly sequentially complete
zbMATH Keywords: Shapiro set
zbMATH Keywords: Bohr compactification
author: Gilles Godefroy
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02772575
OpenAlex ID: W1997853596
description: scientific article; zbMATH DE number 4080120